The easiest bases for the space of nxn matrices is just the matrices with a one as one entry and zeros everywhere else. If you can show that there is a way to make all of these matrice as linear combinations of invertible nxn matrices (just do it explicitly) you are done.
Also remember that the general linear group is not a group of matrices but of isomorphisms of vector spaces. depending on the bases one isomorphism can have different matrices.
